Particle system within particle system?
While particles can give a varied result, it looks like you may prefer a more structured duplication system. I would suggest using dupliverts. Start by creating a mesh that has a vertex positioned where you want each cylinder, only vertices are needed but you can add edges and faces if it makes it easier for you to work with.



Then parent the cylinder to the vertex only mesh, and enable duplication for verts.



You can then repeat the first step with a larger spiral, here I duplicated the first mesh and scaled it up. Then parent the first mesh to this larger one and also enable dupliverts, you will get the first mesh and its cylinders duplicated at each vertex location. You will find it helpful to have the first vertex and the child object both located at the object origin to keep things aligned.